Quandaries
and Queries 

I am sending this for a student because she does not have the Internet. These two problems stumped us and we are going crazy. A bowl contains five green olives and eight black ones. Wilma sticks a fork into a bowl and brings out two olives. What is the probability that she gets one green and one black olive?
What is the probability that there are exactly two girls in a family of three children?
Thank you very much for your time. 

Hi Annetta, I'll do the first one and then you can tell me if it helps you with the second. I would approach both these problems with a tree diagram. For the first problem think of selecting one olive from the jar and then a second. Draw a tree which represents this two step process and write the appropriate probability beside each branch of the tree.In the diagram G represents a green olive and B a black olive. Start at the bottom of the tree. There are 5 green olives and 8 black olives so when you stick the first olive the probability that it is green is ^{5}/_{13} and the probability that it is black is ^{8}/_{13}. Hence I wrote these probabilities beside the bottom two branches of the tree. If you got a green olive then you have climbed the left branch of the tree and the jar has 4 green and 8 black remaining. Stick a second olive. The probability it is green is ^{4}/_{12} and the probability that it is black is ^{8}/_{12}. Thus I labeled the second level of branches, on the left of the tree, with these probabilities. In a similar fashion, if you got a black olive the first time then there are 5 green and 7 black remaining. Hence the probabilities on the right side of the tree. If you have a green olive and a black olive then you got them in one of two ways:
or
The probability that you have a green olive and a black olive is the probability that you selected a green olive and then a black olive or a black olive and then a green olive. This has probability
Harley 

